Physics 111: Mechanics Lecture 14

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Physics 111 Mechanics Lecture 14

• Dale Gary
• NJIT Physics Department

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Life after Phys 111

• The course material of Phys 111 has given you a
  taste of a wide range of topics which are
  available to you as a student. Prerequisite is
  Phys 121 or Phys 121H.
• For those of you who have an interest in
  gravitation/astronomy, I suggest the following
  electives
• Phys 320, 321 Astronomy and Astrophysics I and
  II
• Phys 322 Observational Astronomy
• For those of you interested in the biological or
  BME/medical aspects, I suggest the following
  electives
• Phys 350 Biophysics I, Phys 451 - Biophysics II
• For those of your interested in light, optics,
  and photonics, I suggest the following elective
  which Federici will be teaching this fall and
  Fall 2014
• OPSE 301 Introduction to Optical Science and
  Engineering

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Oscillatory Motion

• Periodic motion
• Spring-mass system
• Differential equation of motion
• Simple Harmonic Motion (SHM)
• Energy of SHM
• Pendulum
• Torsional Pendulum

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Periodic Motion

• Periodic motion is a motion that regularly
  returns to a given position after a fixed time
  interval.
• A particular type of periodic motion is simple
  harmonic motion, which arises when the force
  acting on an object is proportional to the
  position of the object about some equilibrium
  position.
• The motion of an object
• connected to a spring is a
• good example.

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Recall Hookes Law

• Hookes Law states Fs -kx
• Fs is the restoring force.
• It is always directed toward the equilibrium
  position.
• Therefore, it is always opposite the displacement
  from equilibrium.
• k is the force (spring) constant.
• x is the displacement.
• What is the restoring force for a surface water
  wave?

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Restoring Force and the Spring Mass System

• In a, the block is displaced to the right of x
  0.
• The position is positive.
• The restoring force is directed to
• the left (negative).
• In b, the block is at the equilibrium
  position.
• x 0
• The spring is neither stretched nor compressed.
• The force is 0.
• In c, the block is displaced to the left of x
  0.
• The position is negative.
• The restoring force is directed to
• the right (positive).

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Differential Equation of Motion

• Using F ma for the spring, we have
• But recall that acceleration is the second
  derivative of the position
• So this simple force equation is an example of a
  differential equation,
• An object moves in simple harmonic motion
  whenever its acceleration is proportional to its
  position and has the opposite sign to the
  displacement from equilibrium.

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Acceleration

• Note that the acceleration is NOT constant,
  unlike our earlier kinematic equations.
• If the block is released from some position x
  A, then the initial acceleration is kA/m, but
  as it passes through 0 the acceleration falls to
  zero.
• It only continues past its equilibrium point
  because it now has momentum (and kinetic energy)
  that carries it on past x 0.
• The block continues to x A, where its
  acceleration then becomes kA/m.

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Analysis Model, Simple Harmonic Motion

• What are the units of k/m, in
  ?
• They are 1/s2, which we can regard as a
  frequency-squared, so lets write it as
• Then the equation becomes
• A typical way to solve such a differential
  equation is to simply search for a function that
  satisfies the requirement, in this case, that its
  second derivative yields the negative of itself!
  The sine and cosine functions meet these
  requirements.

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SHM Graphical Representation

• A solution to the differential
• equation is
• A, w, f are all constants
• A amplitude (maximum position
• in either positive or negative x direction,
• w angular frequency,
• f phase constant, or initial phase angle.
• A and f are determined by initial conditions.

Remember, the period and frequency are
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Motion Equations for SHM
The velocity is 90o out of phase with the
displacement and the acceleration is 180o out of
phase with the displacement.
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SHM Example 1

• Initial conditions at t 0 are
• x (0) A
• v (0) 0
• This means f 0
• The acceleration reaches extremes
• of w2A at A.
• The velocity reaches extremes of
• wA at x 0.

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SHM Example 2

• Initial conditions at t 0 are
• x (0) 0
• v (0) vi
• This means f - p / 2
• The graph is shifted one-quarter
• cycle to the right compared to the
• graph of x (0) A.

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Consider the Energy of SHM Oscillator

• The spring force is a conservative force, so in a
  frictionless system the energy is constant
• Kinetic energy, as usual, is
• The spring potential energy, as usual, is
• Then the total energy is just

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Transfer of Energy of SHM

• The total energy is contant at all times, and is
  (proportional to the square of
  the amplitude)
• Energy is continuously being transferred between
  potential energy stored in the spring, and the
  kinetic energy of the block.

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Simple Pendulum

• The forces acting on the bob are the tension and
  the weight.
• T is the force exerted by the string
• mg is the gravitational force
• The tangential component of the gravitational
  force is the restoring force.
• Recall that the tangential acceleration is
• This gives another differential equation

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Frequency of Simple Pendulum

• The equation for q is the same form as for the
  spring, with solution
• where now the angular frequency is
• Summary the period and frequency of a simple
  pendulum depend only on the length of the string
  and the acceleration due to gravity. The period
  is independent of mass.

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Torsional Pendulum

• Assume a rigid object is suspended from a wire
  attached at its top to a fixed support.
• The twisted wire exerts a restoring torque on
  the object that is proportional to its angular
  position.
• The restoring torque is t -k q?
• k is the torsion constant of the support wire.
• Newtons Second Law gives

Section 15.5